The Constant of Proportionality

What k is

The constant of proportionality is $y$ divided by $x$: $k = \dfrac{y}{x}$. In a proportional relationship this quotient comes out the same for every pair, and that shared value is $k$.

The order matters. $k$ compares $y$ to $x$, so the $y$-value goes on top. Flipping the division gives $\dfrac{1}{k}$ — a classic wrong answer on multiple choice.

Finding k from a table

Divide $y$ by $x$ in each row. If every row gives the same number, the table is proportional and that number is $k$. If even one row disagrees, the relationship is not proportional and there is no single $k$.

In the table below, every row gives $y \div x = 5$, so the relationship is proportional with $k = 5$ and equation $y = 5x$.

Checking more than one row is not optional — one row can look fine while another breaks the pattern. Two or three quick divisions settle it.

Finding k from a graph

A proportional graph is a straight line through the origin. Pick any clearly marked grid point on the line and divide: $k = \dfrac{y}{x}$. On the graph below the line passes through $(3, 6)$, so $k = \dfrac{6}{3} = 2$ and the equation is $y = 2x$.

There is a shortcut point worth knowing: at $x = 1$, the $y$-value of the line is $k$ itself. That is also why $k$ is called the unit rate — it is how much $y$ you get for one unit of $x$.

Worked examples

Try one yourself

Common questions